You can plot in 3 dimensions only real and/or imaginary parts of a function. One can make use of Plot3D, but since there was a question how the sine function looks like on the unit circle, first I demonstrate usage of ParametricPlot3D and later I'll show a few of many possible uses of Plot3D.

When we'd like to use ParametricPlot3D, then instead of parametrizing complex numbers like x + I y we would rather parametrize them like r * Exp[ I u], where r is a radius of a circle and u is a polar angle. On a unit circle this reduces to Exp[ I u].

The "deepseacolors" and "rainbow" families of curves are respectively parametric 3D - plots of real and imaginary parts of Sine over circles of radius r in the complex plane and the view point rotates around z - axis. The dashed circles are unit circles in planes {x, y} for z in {-1, 0 , 1}. Here the rotation is surplus but still advantageous for the sake of comprehensible visualization.

For a different visualization, on the Riemann sphere, see the paper "Visualizing Complex Functions with the Presentations Application," The Mathematica Journal, vol. 11 #2 (2009), by David J. M. Park and me. Available in CDF and PDF here. (To re-evaluate most code, and to run the dynamic examples, you'll need a copy of Park's Presentations application.)
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murrayAug 7 '12 at 16:57

@murray Thank you for pointing out interesting references. I've seen that paper.
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ArtesAug 7 '12 at 17:01

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